At this stage, the coupling \(\alpha \)  in Eq. (REF ) is dependent on the renormalization scheme used, with the scheme being parametrized by coefficients \(\beta _i\)  (\(i\ge 2\) ) in Eq. (REF ), with \(\partial \alpha /\partial \beta _i\)  depending on \(\beta _i\)  [1]}. As a result, it is not feasible to effect the sort of summation that leads from Eq. (REF ) to Eq. (REF ), so the explicit and implicit dependence of \(R\)  on \(\beta _i\)  cannot be shown to explicitly cancel. However, in Refs. [2]}, [3]}, it is shown that if one replaces the coupling \(\alpha \)  by a parameter \(z\)  which is effectively the 't Hooft coupling, renormalization scheme dependence on the parameters \(\beta _i\)  (\(i\ge 2\) ) is replaced by dependence on a single parameter \(\gamma \) . Again, the RG equation can be used to show that all dependence on \(\gamma \)  cancels in \(R\) , just as dependence of \(R\)  on \(\mu \)  cancels. In addition, the expansion parameters \(\tau _n\)  in Eq. (REF ) can be explicitly shown to be renormalization scheme independent [4]}. This is why we focus on the 't Hooft coupling when considering the analytic running coupling in section .
